What is image registration
- ‘image matching’
- images of same object taken at different times using different methods
- necessary to compare two objects
- determine unknown geometric transformation (rotated, translated, elastically deformed)
Why use image registration
- analyse differences
- combine informations
registration framework
- transformation (mapping)
- similarity measure
optimization
Registration in a closed loop:
1 init 2 estimate similarity 3 check convergence 4 update transformation 5 warp template image -> 2
transformation
goal: bring template image with transformation into alignment with reference image
types of transformation
- highly dependend on expected spatial changes
- affine-linear
- non-linar
Affine linear
- translation only
- rigid transformation (retains size and angular relations; involves translation and rotation)
- scaling (addition of isotropic scale to the rigid transformation)
- affine (preserves lines and parallel lines; angular relations no longer conserved; used when little information about image distortion is known)
Non-linear (deformable) transformations
- no matrix representation, can be very complex
- compute image deformations from a fixed number of basis functions defined by the transformation parameters
- m=3 cubic splines (expressed in basis set of polynomials: B-spline)
B-spline transformations
+ easy to implement and low complexity
+ computationally efficient
- no strict physical meaning
- knot positions not necessarly information-rich positions
common applications of non-linear registration
- tissue motion
- deformation compensation
- longitudinal tissue changes
Registration Basis
- extrinsic: (invasive vs. non-invasive)
- intrinsic: uses information available iwithin the images to estimate the transformation (landmark based, surface to surface)
intensity based similarity measures
- calculation directly from voxel values
- SSD (sum of squared differences)
- NCC (normalized correlation measures)
- joint entropy
- NMI (normalized mutual information)
Sum of squared differences
- subtract each tamplate image voxel from spacial corresponding ref image voxal
- used in academic settings
- unreliable for multi-model registration or registration of contrast varying images
Normalized correlation coefficient
- estimates the difference of two statistical distributions in terms of their standard deviations around the mean
- computationally more expensive
- can be used for multimodal registration
Information theoretic measures
- measures joint information content of reference and template images
- measured by entropy
- treat image intensities as random variables
Joint histogram
- joint entropy, calculated in overlapping points of the image
- stuck in local optima
- maximize mutual information (high marginal entropie, low joint entropy)
- how well does one image explains the other?
Optimization of registration
- iterative numerical optimization method
- maximize similarity, minimize error
- problem: getting stuck in local minimum
Optimization - Gradient Descent
- analyze gradient of E
- parameter change proportional to negated gradient
- slow close to minimum
- finds local minimum
Optimization - Conjugate Gradient Method
- parameter estimation considers estimates from previous steps
- faster convergence than gradient descent
Newton Method
- requires finction that is twice differentiable
Optimization without using derivative information
- powell method: perform line search to find optimal parameters
- Nelder-Mead Method: downhill simplex, heuristic search for nonlinear optimization
Deformable Registration
- often non rigid registration (breathing motion)
- parametric approachers: describe transformation with weighted basis function
- optimization: parameter set and similarity measure for optimization
Optical flow
- assumption: given small temporal difference, a moving voxel does not change intensity
- aperture problem: only motion perpendicular to visible structure can be measured
Diffeomorphism
- diffeomorphic transformations: invertable, no foldings, preserve topology, essential vor computational anatomy
